Motivated by the convolutive behavior of the counting function for partitions with designated summands in which all parts are odd, we consider coefficient sequences $(a_n)_{n\ge 0}$ of primitive eta-products that satisfy the generic convolutive property \begin{align*} \sum_{n\ge 0} a_{mn} q^n = \left(\sum_{n\ge 0} a_n q^n\right)^m \end{align*} for a specific positive integer $m$. Given the results of an exhaustive search of the Online Encyclopedia of Integer Sequences for such sequences for $m$ up to $6$, we first focus on the case where $m=2$ with our attention mainly paid to the combinatorics of two $2$-convolutive sequences, featuring bijective proofs for both. For other $2$-convolutive sequences discovered in the OEIS, we apply generating function manipulations to show their convolutivity. We also give two examples of $3$-convolutive sequences. Finally, we discuss other convolutive series that are not eta-products.

翻译：受具有指定加数的奇部分分拆计数函数的卷积性质启发，我们考虑满足以下一般卷积性质的原始η乘积系数序列$(a_n)_{n\ge 0}$：对于特定正整数$m$，有\\begin{align*} \\sum_{n\\ge 0} a_{mn} q^n = \\left(\\sum_{n\\ge 0} a_n q^n\\right)^m \\end{align*}。基于对在线整数序列百科全书（OEIS）中$m$取值至$6$的此类序列的全面检索结果，我们首先聚焦于$m=2$的情形，重点研究两个$2$-卷积序列的组合性质，并给出两者的双射证明。对于OEIS中发现的其他$2$-卷积序列，我们通过生成函数变换证明其卷积性。同时给出两个$3$-卷积序列的示例。最后，我们讨论非η乘积的其他卷积级数。